Number of complex solutions

Question

Given the following equation:

$$ x^{259}=1 $$

$$ x^{413}=1 $$

How many complex solutions for x have?

Thanks

Answer 1

There are seven solutions of the form $e^{i2k\pi/7}$ for $k=0,...,6$, because the common factor of 259 and 413 is 7. Six of them are "complex" and the other one ($k=0$) is just $1$.

Answer 2

If you mean that $x$ must satisfy both equations, note that $\gcd(259,413)=7$.

Prove now:

1. Every $7$th root of $1$ satisfies both equations.
2. Every solution $x$ of both equations must be a $7$th root of $1$. Hint: use Bezout's identity.